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)| false Alichi, Ali, Jaromir Benes, Joshua Felman, Irene Feng, Charles Freedman, Douglas Laxton, Evan Tanner, David Vavra, and Hou Wang, 2015, , Frontiers of Monetary Policymaking: Adding the Exchange Rate as a Tool to Combat Deflationary Risks in the Czech Republic . IMF Working Paper15/74 Washington DC: IMF
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Appendix I. First-Order Conditions for Calvo Price Setting
In the non-linearized model, the Calvo Price setting first-order conditions can be expressed as follow:
with PPI inflation denoted
Appendix II. Derivation of the Intertemporal Budget Constraint
Assuming that foreign countries are symmetric, Home’s budget constraint at date t, which is derived by adding the budget constraints of:
- the households:
- the central bank:
- the government:
The consolidated budget contraint is then:
We define the country net foreign asset position31 and the trade balance as follows:
Introducing the net foreign asset position and the trade balance in the consolidated budget constraint, we have:
The Euler equation for foreign households is:
Using the two above equations, we then have:
Appendix III. Derivation of the Loss Function
We first have the exact relationship:
And the following second-order approximation of the goods market clearing condition Yt = StC*[(l—α)Θt + α:
We use this result to derive:
By the labor market clearing condition, we have up to second-order approximation:
By Woodford (2003), we have:
We now use a second order approximation of the budget constraint to replace the linear term θt in the expression above. We find:
We then get the following loss function (up to additive terms independent of policy and multiplicative constants):
Appendix IV. Proof of Proposition 1
For analytical convenience, we express the consumption distortion θt in deviation from the steady state. The problem boils down to:
Proof. The first-order conditions on the two instruments are:
We are grateful for the comments and suggestions from Suman Basu, Paolo Cavallino, Nicolas Coeurdacier, Emmanuel Farhi, Philippe Martin, Helene Rey, Lars Svensson, Pablo Winant, Felipe Zanna, and from seminar participants at the IMF and at the Bank of England. The views expressed in this paper are solely those of the authors and do not represent those of the IMF or IMF policy.
According to the IMF classification, around one-third of all countries either de jure or de facto manage their exchange rate.
Sterilized intervention consists of the central bank purchasing or selling foreign currency-denominated assets with corresponding sales or purchases of domestic currency assets in order to leave the money supply unchanged. If FX intervention is not sterilized, then it does not constitute a separate instrument from monetary policy.
See Farhi and Werning (2014) or Woodford (2012) for models in which financial frictions justify the use of capital controls or unconventional policies. In practice, FX intervention is the most commonly used tool after the policy interest rate.
Of course, similar stories could be told of central banks that attempted to limit depreciation using FX intervention and eventually had to raise rates. In fact, most central banks have followed such strategies in the major currency crises — e.g. Mexico in 1994; Thailand in 1997; Brazil in 1998; Russia in 1998; etc — albeit with limited success.
The natural rate of interest rate is the rate that would prevail for in the equilibrium with flexible prices.
We allow for a constant labor tax τ to make the steady state efficient.
Ξt could also be thought of as capturing time-varying and country-specific borrowing constraints.
In Gabaix and Maggiori (2015), the effectiveness of FX intervention (captured by our elasticity φ) would depend on the importance of the friction affecting the international financial intermediaries, and on the relation of the size of intervention to the size of the intermediaries’ balance sheet (which would be proxied by our parameter
In reality, households in small open economies do invest part of their net wealth abroad, but they do not run carry trade, i.e., borrow in local currency at a high interest rate to invest abroad at a low rate, even though the local exchange rate may be depreciating faster than what the UIP would imply. The reasons for limited carry trade from emerging economies may be due to liquidity constraints, short-termism (since future capital gains are offset by today’s losses in interest income), or regulatory constraints on FX position. Such regulatory constraints certainly also prevent households to borrow significant amounts in foreign currency to invest at home.
Reserves may be insufficient to offset large expected depreciations, but we do not discuss this possibility here.
For simplicity, we assume that only the Home central bank uses FX intervention. This assumption will be natural in our context of a small open economy taking the rest of the world as given.
Using the definition of Θt, we have
Since firms’ borrowing is domestic, interest payments are collected by domestic investors and thus they do not affect the country’s budget constraint at date t.
We normalize the net foreign asset position by the foreign consumption and take the foreign price at home PF,t as the numeraire.
Note that the model and the loss function are presented differently from that in Farhi and Werning (2014), who present variables in deviation from the natural allocation, whereas we express them in deviation from the deterministic steady state.
Basu, Ghosh, Ostry and Winant (2017) analyze the time consistent equilibrium when reserves are limited and central bank policy is time inconsistent.
This effect exists as long as the economy is not fully open, i.e. as long as the share α of domestic goods consumed by domestic households is not zero.
More precisely, as shown by Farhi and Werning (2014), optimal policy with a fully open economy would consist of maximizing the monopoly profits of exporters, but given that under the Cole-Obstfeld parameterization, exporters face a demand with elasticity of 1, the monopoly problem is degenerate. As a result, output should converge to 0 when α → 1, and distorting consumption does not help with the terms of trade, which is why FX intervention is useless.
The quadratic approximation of the budget constraint included in the welfare objective does not ensure that this constraint is satisfied; the presence of the intertemporal constraint in the objective function is necessary (due to the microfoundations) but not sufficient to ensure that the constraint is satisfied.
When the financial accelerator is active (i.e. μ ≠ 0)), the results become:
This effect is visible through the domestic consumption distortion that results from the exogenous risk premium shock
Adding a constraint that
In the case where D(αθ, •) = +∞, Proposition 1 shows that output and inflation are 0.
Since there are no exogenous risk premium shock in this section, the total consumption distortion θt coincides with the consumption distortion resulting from foreign exchange interventions
More precisely, inflation is always lower than future inflation in the absence of a financial accelerator; thus, the only non-explosive path for inflation (and the exchange rate) is the one with zero inflation. This is why when the Blanchard-Khan condition is always satisfied in this case.
Normalizing by foreign consumption and taking the foreign price at home PF,t as the numeraire.